induced by this. Every such cocycle ∇∈Hdiff(X,BnA)\nabla \in \mathbf{H}_{diff}(X,\mathbf{B}^n A) we may think of as an ∞-connection on the Bn−1A\mathbf{B}^{n-1}A-principal ∞-bundle classified by the underlying cocycle in H(X,BnA)\mathbf{H}(X, \mathbf{B}^n A).

Notice that the complex of sheaves ♭BnU(1)\mathbf{\flat}\mathbf{B}^n U(1) is that which defines flatDeligne cohomology, while that of ♭dRBnU(1)\mathbf{\flat}_{dR} \mathbf{B}^n U(1) is essentially that which defines de Rham cohomology in degree n>1n \gt 1 (see below). Also notice that we denoted by ddRd_{dR} also the differential C∞(−,U(1))→ddRlogΩ1(−)C^\infty(-,U(1)) \stackrel{d_{dR} log}{\to} \Omega^1(-); this is to stress that we are looking at U(1)U(1) as the quotient ℝ/ℤ\mathbb{R}/\mathbb{Z}.

Proof

Since the global section functor Γ\Gamma amounts to evaluation on the point ℝ0\mathbb{R}^0 and since constant simplicial presheaves on CartSp satisfy descent (on objects in CartSpCartSp!), we have that ♭BnU(1)\mathbf{\flat} \mathbf{B}^n U(1) is represented by the complex of sheaves Ξ[constU(1)→0→⋯→0]\Xi[const U(1) \to 0 \to \cdots \to 0]. This is weakly equivalent to Ξ[C∞(−,U(1))→ddRΩ1(−)→ddR⋯→ddRΩcln(−)]\Xi[C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n_{cl}(-)] by the Poincare lemma applied to each Cartesian space (using the same standard logic that proves the de Rham theorem) in that the degreewise inclusion

is the pullback over a cospan all whose objects are fibrant and one of whose morphisms is a fibration. Therefore this is a homotopy pullback diagram in [CartSpop,sSet]proj[CartSp^{op}, sSet]_{proj} which models the (∞,1)-limit over *→BnU(1)←♭BnU(1)* \to \mathbf{B}^n U(1) \leftarrow \mathbf{\flat}\mathbf{B}^n U(1) in PSh(∞,1)(CartSp)PSh_{(\infty,1)}(CartSp). Since ∞-stackification preserves finite (∞,1)(\infty,1)-limits this models also the corresponding (∞,1)(\infty,1)-limit in H\mathbf{H}. Therefore the top left object is indeed a model for ♭dRBnU(1)\mathbf{\flat}_{dR} \mathbf{B}^n U(1).

Proof

Let {Ui→X}\{U_i \to X\} be a good open cover. At Smooth∞Grpd is discussed that then the Cech nerveC({Ui})→XC(\{U_i\}) \to X is a cofibrant resolution of XX in [CartSpop,sSet]proj,cov[CartSp^{op}, sSet]_{proj,cov}. Therefore we have

of differential forms, with Ci∈Ωcln(Ui)C_i \in \Omega^n_{cl}(U_i), Bij∈Ωn−1(Ui∩Uj)B_{i j} \in \Omega^{n-1}(U_i \cap U_j), etc. , such that this collection is annihilated by the total differentoal D=ddR±δD = d_{dR} \pm \delta, where ddRd_{dR} is the de Rham differential and δ\delta the alternating sum of the pullbacks along the face maps of the Cech nerve.

By recurseively adding coboundaries this way, we can annihilate all the higher Cech-components of the original cocycle and arrive at a cocycle of the form (Fi,0,⋯,0)(F_i, 0, \cdots, 0).

Such a cocycle being DD-closed says precisely that Fi=F|UiF_i = F|_{U_i} for F∈Ωcln(X)F \in \Omega^n_{cl}(X) a globally defined closed differential form. Moreover, coboundaries between two cocycles both of this form

Here on the right we have the subset of Deligne cocycles that picks for each integral de Rham cohomology class of XX only one curvature form representative.

We give the proof below after some preliminary expositional discussion.

Remark

The restriction to single representatives in each de Rham class is a reflection of the fact that in the above (∞,1)(\infty,1)-pullback diagram the morphism HdR(X,Bn+1U(1))→HdR(X,Bn+1U(1))H_{dR}(X,\mathbf{B}^{n+1}U(1)) \to \mathbf{H}_{dR}(X,\mathbf{B}^{n+1}U(1)) by definition picks one representative in each connected component. Using the above model of the intrinsic de Rham cohomology in terms of globally defined differential froms, we could easily get rid of this restriction by considering instead of the above (∞,1)(\infty,1)-pullback the homotopy pullback

where now the right vertical morphism is the inclusion of the set of objects of our concrete model for the ∞\infty-groupoid HdR(X,Bn+1U(1))\mathbf{H}_{dR}(X, \mathbf{B}^{n+1} U(1)). With this definition we get the isomorphism

From the tradtional point of view of differential cohomology this may be what one expects to see, but from the intrinsic (∞,1)(\infty,1)-topos theoretic point of view it is quite unnatural – and in fact “evil” – to fix that set of objects of the ∞\infty-groupoid. Of intrinsic meaning is only the set of their equivalences classes.

Circle bundles with connection

This contains in it already all the relevant structure of the general case, but the low categorical degree is more transparently written out and will allow us to pause to highlight some maybe noteworthy aspects of the situation, such as the phenomenon of pseudo-connectionsbelow.

In terms of the Dold-Kan correspondence the object BU(1)∈H\mathbf{B}U(1) \in \mathbf{H} is modeled in [CartSpop,sSet][CartSp^{op}, sSet] by

In order to compute the differential cohomology Hdiff(−,BU(1))\mathbf{H}_{diff}(-,\mathbf{B}U(1)) by an ordinary pullback in sSet we also want to resolve the curvature characteristic morphism BU(1)→♭dRB2U(1)\mathbf{B}U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^2 U(1) by a fibration. We claim that this may be obtained by choosing the resolution BU(1)←≃BU(1)diff,chn\mathbf{B}U(1) \stackrel{\simeq}{\leftarrow} \mathbf{B} U(1)_{diff,chn} given by

For XX a smooth manifold, morphisms in [CartSpop,2Grpd][CartSp^{op}, 2Grpd] of the form traA:Π2(X)→EBU(1)tra_A : \Pi_2(X) \to \mathbf{E}\mathbf{B}U(1) are in bijection with smooth 1-forms A∈Ω1(X)A \in \Omega^1(X): the 2-functor sends a path in XX to the the parallel transport of AA along that path, and sends a surface in XX to the exponentiated integral of the curvature 2-form FA=dAF_A = d A over that surface. The Bianchi identitydFA=0d F_A = 0 says precisely that this assignment indeed descends to homotopy classes of surfaces, which are the 2-morphisms in Π2(X)\Pi_2(X).

Moreover 2-morphisms of the form (λ,α):traA→traA′(\lambda,\alpha) : tra_A \to \tra_{A'} in [CartSpop,2Grpd][CartSp^{op}, 2Grpd] are in bijection with pairs consisting of a λ∈C∞(X,U(1))\lambda \in C^\infty(X,U(1)) and a 1-form α∈Ω1(X)\alpha \in \Omega^1(X) such that A′=A+ddRλ−αA' = A + d_{dR} \lambda - \alpha.

So by the above definition of differential cohomology in H\mathbf{H} we find that BU(1)\mathbf{B}U(1)-differential cohomology of a paracompactsmooth manifoldXX is given by choosing any good open cover{Ui→X}\{U_i \to X\}, taking C({Ui})C(\{U_i\}) to be the Cech nerve, which is then a cofibrant replacement of XX in [CartSpop,sSet]proj,cov[CartSp^{op}, sSet]_{proj,cov} and forming the ordinary pullback

(because the bottom vertical morphism is a fibration, by the fact that our model for BdiffU(1)→♭dRB2U(1)\mathbf{B}_{diff} U(1) \to \flat_{dR}\mathbf{B}^2 U(1) is a fibration, that C({Ui})C(\{U_i\}) is cofibrant and using the axioms of the sSet-enriched model category[CartSpop,sSet]proj[CartSp^{op}, sSet]_{proj}).

Observations

A cocycle in [CartSpop,sSet](C({Ui}),BdiffU(1))[CartSp^{op},sSet](C(\{U_i\}), \mathbf{B}_{diff}U(1)) is

in the above model[CartSpop,sSet](C({Ui}),♭dRB2U(1))[CartSp^{op},sSet](C(\{U_i\}), \mathbf{\flat}_{dR}\mathbf{B}^2 U(1)) for intrinsic de Rham cohomology.

Every cocycle with nonvanishing (aij)(a_{i j}) is in [C({Ui}),BdiffU(1)][C(\{U_i\}), \mathbf{B}_{diff}U(1)] coboundant to one with vanishing (aij)(a_{i j})

Proof

The first statements are effectively the definition and the construction of the above models. The last statement is as in the above discussion of our model for ordinary de Rham cohomology: given a cocycle with non-vanishing closed aija_{i j}, pick a partition of unity (ρi∈C∞(X))(\rho_i \in C^\infty(X)) subordinate to the chosen cover and the coboundary given by (∑i0ρi0ai0i)(\sum_{i_0} \rho_{i_0} a_{i_0 i}). This connects (Ai,aij,gij)(A_i,a_{i j}, g_{i j}) with the cocycle (A′i,a′ij,gij)(A'_i, a'_{i j}, g_{i j}) where

and we find that it lives in the sheaf hypercohomology that models ordinary de Rham cohomology.

Therefore we find that in each cohomology class of curvatures, there is at least one representative which is an ordinary globally defined 2-form. Moreover, the pseudo-connections that map to such a representative are precisely the genuine connections, those for which the (aij)(a_{i j})-part of the cocycle vaishes.

So we see that ordinary connections on ordinary circle bundles are a means to model the homotopy pullback

in a 2-step process: first the choice of a pseudo-connection realizes the bottom horizontal morphism as an anafunctor, and then second the restriction imposed by forming the ordinary pullback chooses from all pseudo-connections precisely the genuine connections.

Circle bundles with pseudo-connection

In the above discussion of extracting ordinary connections on ordinary U(1)U(1)-principal bundles from the abstract topos-theoretic definition of differential cohomology, we argued that a certain homotopy pullback may be computed by choosing in the Cech-hypercohomology of the complex of sheaves (Ω1(−)→ddRΩcl2(−))(\Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2_{cl}(-)) over a manifold XX those cohomology representatives that happen to be represented by globally defined 2-forms on XX. We saw that the homotopy fiber of pseudo-connections over these 2-forms happened to have connected components indexed by genuine connections.

But by the general abstract theory, up to isomorphism the differential cohomology computed this way is guaranteed to be independent of all such choices, which only help us to compute things.

To get a feeling for what is going on, it may therefore be useful to re-tell the analgous story with pseudo-connections that are not genuine connections.

By the very fact that BU(1)←≃BdiffU(1)\mathbf{B}U(1) \stackrel{\simeq}{\leftarrow} \mathbf{B}_{diff}U(1) is a weak equivalence, it follows that every pseudo-connection is equivalent to an ordinary connection as cocoycles in [CartSpop,sSet](C({Ui}),Bdiff(G))[CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}_{diff}(G)).

If we choose a partition of unity(ρi∈C∞(X,ℝ))(\rho_i \in C^\infty(X,\mathbb{R})) subordinate to the cover {Ui→X}\{U_i \to X\}, then we can construct the corresponding coboundary explicitly:

is one which takes all the ordinary curvature forms to vanish identically

∇=(Ai:=0,gij,aij).
\nabla = (A_i := 0, g_{i j}, a_{i j})
\,.

This fixes the pseudo-components to be aij=−dgija_{i j} = - d g_{i j}. By the above discussion, this pseudo-connection with vanishing connection 1-forms is equivalent, as a pseudo-connection, to the ordinary connection cocycle with connection forms (Ai:=∑i0ρi0dgi0i)(A_i := \sum_{i_0} \rho_{i_0} d g_{i_0 i}). This is a standard formula for equipping U(1)U(1)-principal bundles with Cech cocycle (gij)(g_{i j}) with a connection.

U(1)0U(1)_0-groupoid bundles

We saw above that the intrinsic coefficient object ♭dRBnU(1)\mathbf{\flat}_{dR} \mathbf{B}^n U(1) yields ordinary de Rham cohomology in degree n>1n \gt 1. For n=1n = 1 we have that ♭dRBU(1)\mathbf{\flat}_{dR} \mathbf{B}U(1) is given simply by the 0-truncated sheaf of 1-forms, Ω1(−):CartSpop→Set↪sSet\Omega^1(-) : CartSp^{op} \to Set \hookrightarrow sSet. Accordingly we have for XX a paracompact smooth manifold

For X→Bn−1U(1)X \to \mathbf{B}^{n-1} U(1) a cocycle (an (n−2)(n-2)-gerbe without connection), the cohomology class of the composite X→Bn−1U(1)→♭dRBnU(1)X \to \mathbf{B}^{n-1} U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^n U(1) is precisely the obstruction to the existence of a flat extension X→♭Bn−1U(1)→Bn−1U(1)X \to \mathbf{\flat} \mathbf{B}^{n-1} U(1) \to \mathbf{B}^{n-1} U(1) for the original cocycle.

For n=2n = 2 this is the usual curvature 2-form of a line bundle, for n=3n = 3 it is curvature 3-form of a bundle gerbe, etc. But for n=1n = 1 we have that the original cocycle is just a map of spaces

f:X→U(1).
f : X \to U(1)
\,.

This can be understoody as a cocycle for a groupoid principal bundle, for the 0-truncated groupoid with U(1)U(1) as its space of objects. Such a cocycle extends to a flat cocycle precisely if ff is constant as a function. The corresponding curvature 1-form is ddRfd_{dR} f and this is precisely the obstruction to constancy of ff already, in that ff is constant if and only if ddRfd_{dR} f vanishes. Not (necessarily) if it vanishes in de Rham cohomology .

This is the simplest example of a general statement about curvatures of higher bundles: the curvature 1-form is not subject to gauge transformations.

Therefore this is a homotopy pullback in [CartSpop,sSet]proj[CartSp^{op}, sSet]_{proj} that realizes the (∞,1)(\infty,1)-pullback in question in the (∞,1)-category of (∞,1)-presheavesPSh(∞,1)(CartSp)PSh_{(\infty,1)}(CartSp). Since ∞-stackification preserves finite (∞,1)-limits, it therefore also presents the above (∞,1)(\infty,1)-pullback in H=Sh(∞,1)(CartSp)\mathbf{H} = Sh_{(\infty,1)}(CartSp).

Proof

For the lower square we had discussed this already above. For the upper square the same type of reasoning applies. The main point is to find the chain complex in the top right such that it is a resolution of the point and maps by a fibration onto our model for ♭BnU(1)\mathbf{\flat}\mathbf{B}^n U(1). The top right complex is

and the vertical map out of it into C∞(−,U(1))→ddRΩ1(−)→⋯→ddRΩn(−)→ddRΩcln+1(−)C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n(-) \stackrel{d_{dR}}{\to} \Omega^{n+1}_{cl}(-) is in positive degree the projection onto the lower row and in degree 0 the de Rham differential. This is manifestly surjective (by the Poincare lemma applied to each object U∈U \in CartSp) hence this is a fibration.

and in turn the top left vertical morphism curv:BdiffnU(1)→♭dRBn+1U(1)curv : \mathbf{B}_{diff}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1) is in positive degree the projection on the lower row and in degree 0 the de Rham differential.

is a Kan fibration by the fact that [CartSpop,sSet]proj[CartSp^{op}, sSet]_{proj} is an sSetQuillensSet_{Quillen}-enriched model category. Therefore the homotopy pullback is computed as an ordinary pullback.

By the above discussion of de Rham cohomology we have that we can assume the morphism HdRn+1(X)→[CartSpop,sSet](C({Ui}),♭dRBn+1)H_{dR}^{n+1}(X) \to [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{\flat}_{dR}\mathbf{B}^{n+1}) picks only cocylces represented by globally defined closed differential forms F∈Ωn+1(X)F \in \Omega^{n+1}(X).

By the nature of the chain complexes apearing in the above proof, we see that the elements inm the fiber over such a globally defined form are precisely the cocycles with values only in the “upper row complex”

in [CartSpop,sSet][CartSp^{op}, sSet] for the canoncal curvature characteristic class curv:BnU(1)→♭dRBn+1U(1)curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1) in Smooth∞Grpd with the special property that it did model the abstract (∞,1)-topos-theoretic class under the Dold-Kan correspondence precisely in terms of the familiar Deligne cohomology coefficient complex.

For distinguishing the two models, we will indicate the former one by the subscript chn{}_{chn} and the one described now by the subscript simp{}_{simp}.

Convention

Here and in the following we adopt for differential forms on simplices the following notational convention:

by Ω•(Δn)\Omega^\bullet(\Delta^n) we denote the complex of smooth differential forms on the standard smooth nn-simplex with sitting instants: for every k∈ℕk \in \mathbb{N} every kk-face of Δn\Delta^n has a neighbourhood of its boundary such that the form restricted to that neighbourhood is constant in the direction perpendicular to that boundary.

Here CE(bn−1ℝ)CE(b^{n-1}\mathbb{R}) is the Chevalley-Eilenberg algebra of bn−1ℝb^{n-1}\mathbb{R}, which is simply the graded-commutative dg-algebra (over ℝ\mathbb{R}) on a single generator in degree nn with vanishing differential.

Moreover, coskn+1(−)\mathbf{cosk}_{n+1}(-) is the coskeleton-operation and the quotient is by constant nn-forms ω∈Ωcln(U×Δk)vert\omega \in \Omega^n_{cl}(U \times \Delta^k)_{vert} such that ∫Δnω∈ℤ\int_{\Delta^n}\omega \in \mathbb{Z}. We take the quotient as a quotient of abelian simplicial groups (the group operation is the addition of differential forms).

Remark

The set of square diagrams of dg-algebras above is over (U,[k])(U,[k]) the set of nn-forms ω\omega on U×ΔkU \times \Delta^k whose curvature(n+1)(n+1)-form dωd \omega has no component with all legs along Δk\Delta^k.

Proposition

The morphism given by fiber integration of differential forms over the simplex factor fits into a diagram

Proposition

Proof

Observe that Bnℝdiff,simp\mathbf{B}^n \mathbb{R}_{diff,simp} is the pullback of ♭dRBn+1ℝsimp→♭Bn+1ℝsimp\mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R}_{simp} \to \mathbf{\flat}\mathbf{B}^{n+1} \mathbb{R}_{simp} along the evident forgetful morphism from

This forgetful morphism is evidently a fibration (because it is a degreewise surjection under Dold-Kan), hence this pullback models the homotopy fiber of ♭dRBn+1ℝ→♭Bn+1ℝ\mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R} \to \mathbf{\flat} \mathbf{B}^{n+1} \mathbb{R}. Since by the above fiber integration gives a weak equivalence of pulback diagrams the claim follows.

Definition

Write BnU(1)conn,simp↪BnU(1)diff,simp\mathbf{B}^n U(1)_{conn,simp} \hookrightarrow \mathbf{B}^n U(1)_{diff,simp} for the sub-presheaf which over (U,[k])(U,[k]) is the set of those forms ω\omega on U×ΔkU \times \Delta^k such that the curvaturedωd \omega has no leg along Δk\Delta^k.

In summary this gives us the following alternative perspective on connections on Bn−1U(1)\mathbf{B}^{n-1}U(1)-principal ∞-bundles: such a connection is a cocycle with values in the Bnℤ\mathbf{B}^n \mathbb{Z}-quotient of the (n+1)(n+1)-coskeleton of the simplicial presheaf which over (U,[k])(U,[k]) is the set of diagrams of dg-algebras

See the discussion at homotopy pullback for why this is indeed interpreted by the homotopy pullback BnU(1)×♭dRBn+1U(1)Ωcln+1\mathbf{B}^n U(1) \times_{\mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)} \Omega^{n+1}_{cl}.

Examples

For n=1n = 1 a circle nn-bundle with connection in the sense discussed here is indeed an ordinary hermitian line bundle or equivalently U(1)U(1)-principal bundle with connection.

For n=2n = 2 a circle 2-bundle with connection is equivalent to a bundle gerbe with connection (at least over a smooth manifold. Over an orbifold the definition given here does produce the correct equivariant cohomology, which is different from that of bundle gerbes that are equivariant in the ordinary sense.)

Classes of examples of higher circle bundles with connection are provided by ∞-Chern-Weil theory which provides homomorphisms of the form